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Money and Modern Banking without Bank Runs
David R. Skeie�
Federal Reserve Bank of New York
October 2006
Abstract
Bank runs in the literature take the form of withdrawals of demand deposits
payable in real goods, which deplete a �xed reserve of goods in the banking system.
This framework describes traditional bank runs based on currency withdrawals as oc-
curred historically in the U.S. and more recently in developing countries. However,
in a modern banking system, large withdrawals typically take the form of electronic
payments within a clearinghouse system of banks. These transfers shift balances
among banks, with no analog of a depletion of a scarce reserve from the banking
system. A new framework of nominal demand deposits repayable in money within a
clearinghouse can examine bank run threats in a modern developed economy. This ap-
proach shows that interbank lending and monetary prices may be able to prevent pure
liquidity-driven bank runs, demonstrating the resiliency of modern banking systems
to excessive withdrawals. This �nding suggests that the rationalization for deposit
insurance based on the Diamond-Dybvig model of real demand deposits holds for
developing countries but not necessarily for developed countries.
JEL Classi�cation: G21, G28, E42
Keywords: Bank runs, nominal contracts, demand deposits, interbank market, clear-
inghouse, payments, deposit insurance, inside money�Email: [email protected]. This paper is a revised portion of a previously circulated paper, �Money
and Modern Bank Runs,�which is the �rst chapter of my dissertation thesis at Princeton University. Iam grateful to my advisors Franklin Allen, Ben Bernanke and Patrick Bolton, and to Ken Ayotte, MarkusBrunnermeier, Ed Green, Yair Listokin, Guido Lorenzoni, Antoine Martin, Jamie McAndrews, Cyril Mon-net, Wei Xiong, seminar participants at Federal Reserve Bank of Chicago, Federal Reserve Bank of NewYork, Federal Reserve Bank of Richmond, Federal Reserve Board of Governors, FDIC, NYU Stern, Prince-ton, UNC Chapel Hill Kenan-Flager, Wharton, Sveriges Riksbank Workshop on Banking, Financial Stabil-ity and the Business Cycle, 2004 SED Annual Meeting (Florence), 2005 FMA Annual Meeting (Chicago),JBF 30th Anniversary Conference (Beijing) and FIRS Conference on Banking, Corporate Finance andIntermediation (Shanghai) for helpful comments and conversations. The views expressed in this paper arethose of the author and do not necessarily re�ect the views of the Federal Reserve Bank of New York orthe Federal Reserve System. All errors are mine.
1 Introduction
The number of countries with deposit insurance has more than quadrupled to 87 since
the classic paper by Diamond and Dybvig (1983) was published.1 Their paper serves
as a major theoretical justi�cation for deposit insurance by claiming that it prevents
otherwise inevitable liquidity-based bank runs. However, some evidence suggests that
deposit insurance may hurt more than help. Deposit insurance may reduce monitoring of
banks and increase moral hazard and portfolio volatility,2 leading to a greater probability
of bank failures,3 with the S&L crisis in the 1980s as a leading example.4 Several authors
argue that deposit insurance is adopted by countries based on rent-seeking private interests
due to political power.5 While developing countries have recently increasingly adopted
deposit insurance, developed countries have been the primary adopters overall, and have
been particularly linked to the private-interests theory.6
This paper studies how the threat of bank runs in developed countries di¤ers from
that in developing countries, which has implications for the Diamond-Dybvig justi�cation
for deposit insurance in each case. To re�ect the modern economy and banking system
of developed countries, as de�ned below, I introduce a new framework of nominal mon-
etary demand deposits into a Diamond-Dybvig setting, in which banks pay withdrawals
in monetary payments through a clearinghouse system of banks. In this framework, in-
terbank lending and a monetary-priced market for goods may be able to prevent pure
liquidity-driven bank runs that occur in Diamond-Dybvig.
A nominal deposits framework contrasts with the standard real deposits framework
in current theory. Starting with Diamond-Dybvig, banks are modeled in the literature as
paying withdrawals of deposits in real goods. The excessive early withdrawals of deposits
are initially triggered due to various causes in the di¤erent strands of the literature, includ-
ing multiple equilibria following Diamond-Dybvig and asymmetric information regarding
asset shocks following Chari and Jagannathan (1988) and Calomiris and Kahn (1991). In
all of these strands of the general real deposits framework, excessive withdrawals deplete
a �xed reserve of liquid real goods available to be paid out from the banking system. Be-
1Demirguc-Kunt (2005).2Kim and Santomero (1988), Martin (2006), Nagarajan and Sealey (1995) and Penati and Protopa-
padakis (1988).3Demirguc-Kunt and Detragiache (1997, 2002).4Kane (1989).5Kroszner (1998), Kroszner and Strahan (2001) and Laeven (2004).6Demirguc-Kunt et al. (2005).
1
cause demand deposits are modeled as �xed promises of goods, payments to withdrawers
cannot be rationed. Long term investments have to be ine¢ ciently liquidated to provide
short term payouts. The bank will not be able to pay future withdrawals, so all depositors
try to withdraw immediately. This bank run is inevitable in the real deposits framework
due to the fragile liquidity structure of the bank�s real short term liabilities versus real
long term assets, even though the bank may be otherwise fundamentally solvent. Deposit
insurance provides guarantees to depositors on their withdrawals, so that they do not run
the bank.
The real deposit models of bank runs describe traditional depositor runs that typically
occur in developing countries and that occurred historically in the U.S. during the 19th and
early 20th century. Gorton (1988) and Friedman and Schwartz (1963) show that during
banking crises in this era, the ratio of currency to deposits increased. This implies that
depositors withdrew currency and stored it outside of the banking system, corresponding
to bank run models in which real goods are withdrawn from the banking system. Allen
and Gale (1998) and Smith (2003) cite the withdrawals of currency from the banking
system during these historical bank runs as what their models explain. Diamond and
Rajan (2005) state their real deposit model most closely resembles the gold standard era
in the U.S., with an in�exible value of money, and the recent banking crisis in Argentina,
with deposits repayable in dollars that could be withdrawn from the country.
The major threat from excessive withdrawals is di¤erent in a modern economy and
banking system, de�ned as one with a �oating currency, bank deposits denominated in
domestic currency, and a modern clearinghouse payments system. Large bank withdrawals
typically take the form of intraday electronic transfers of money between banks within a
clearinghouse.7 While money balances shift among banks, there is no correspondence to
the real-deposits bank run literature of a depletion of a scarce reserve. An example is a
wholesale depositor who does not roll over large CDs at a bank and redeposits the funds
elsewhere, or withdraws to make purchases. Regardless, the money from the wholesaler�s
account at his current bank is sent to either his new account at a di¤erent bank or to the7An electronic withdrawal is much more practical and timely than a costly and risky physical withdrawal
of a large sum of currency, especially if there is an imminent run on a bank. Demirguc-Kunt et al. (2004)show that in contemporary times, aggregate bank deposits do not signi�cantly decline during times of�nancial distress, especially in developed countries and even in many less-developed economies. Thisstrongly suggests that currency is not withdrawn from the banking system in any critical amount inmodern economies. Skeie (2004) examines nominal deposits allowing for withdrawals of currency, whichmay be stored outside of the banking system, in addition to withdrawals in the form of banking paymentswithin a clearinghouse considered here.
2
bank of the party who is selling to the wholesaler. The receiving bank may then lend out
or pay these funds yet again to other banks during the same day.
A new approach is needed to examine the threat of bank runs in a modern economy.
I develop a model of nominal demand deposits that are withdrawn by depositors as elec-
tronic payments within a clearinghouse. This approach highlights how a modern banking
system is resilient to excessive withdrawals in two ways. First, it shows that fundamen-
tally, the interbank market always has available any amount of funds that are demanded
for withdrawal and interbank loans. If all depositors withdraw from a bank, all funds are
paid to a second bank for either redeposits or purchases. The second bank always has
the capability of lending back to the �rst bank. Moreover, it prefers to lend back if the
run is triggered for pure liquidity reasons, as in Diamond-Dybvig. Lending to prevent the
liquidation of the �rst bank�s current long term investments is naturally most e¢ cient and
pays the greatest return on funds. Thus, the �rst bank does not fail, even with excess
early withdrawals, so pure liquidity-driven runs do not occur in equilibrium in the �rst
place.
In contrast, the literature shows that by design, when demand deposits are repayable in
goods, interbank lending cannot prevent large enough pure liquidity runs.8 If the aggregate
amount of early withdrawals is greater than the total amount of liquid goods held by all
banks, there are not enough liquid goods available to lend and runs occur. This paper
points out that unless currency is withdrawn and stored outside of the banking system,
bank reserves are not drained from the banking system in a closed economy absent central
bank intervention, and monetary aggregates do not change for a given money multiplier.
When deposits are repayable in monetary claims within a clearinghouse, an interbank
market for money has the capacity to lend to a bank in need regardless of the amount of
withdrawals, so interbank lending can in principle prevent liquidity runs.
Second, the approach in this paper shows that nominal deposits hedge banks individ-
ually or in aggregate against excessive early withdrawals. Banks initially lend money to
entrepreneurs who store and invest goods and then sell them on the current goods mar-
ket. If depositors run a bank by making excessive early withdrawals to purchase goods,
an abundance of money to buy goods drives the price up in the early period. The bank�s
short term liabilities increase in nominal but not real value, since deposits are nominal and
8Bhattacharya and Gale (1987), Bhattacharya and Fulghieri (1994), Allen and Gale (2000a, 2004),Aghion et al. (2000) and Diamond and Rajan (2005).
3
the price level increases. In real value terms, the bank�s short term assets decrease but
its long term assets increase, leaving it fundamentally solvent. The bank can borrow back
from the interbank market the excess funds it paid out, requiring no liquidation of long
term assets. Higher prices in the goods market ration consumption to early purchasers
and save goods for those who do not run the bank. Depositors face lower relative prices
at the later period and prefer to purchase goods then, so a run never occurs in equilib-
rium. Moreover, the price mechanism in the goods market provides the (ex-ante) optimal
allocation of goods among consumers.
Several papers in the literature examine reduced form or partial equilibrium models
of money for deposit withdrawals and interbank or central bank loans.9 However, these
are models of real, not nominal, demand deposits. Money paid for deposit withdrawals
is either consumed or withdrawn from the economy, with no market for goods and no
endogenous monetary price level.10 I show that with a general equilibrium model of money,
monetary prices imply nominal deposits hedge banks when there are excess withdrawals
by depositors purchasing goods.
An implication from the nominal deposits framework is that Diamond-Dybvig does not
necessarily justify deposit insurance for preventing pure liquidity-driven runs in developed
countries. In contrast, the justi�cation for deposit insurance based on Diamond-Dybvig
is clari�ed for developing countries with non-modern economies and banking systems.
Deposit insurance is needed in these countries where currency typically has a �xed value
and is often withdrawn and held outside of the banking system.
This paper does not claim that bank runs cannot occur in a modern economy. Rather,
the focus of this paper is to show the fundamental di¤erences between nominal and real
deposits frameworks by using a simple, frictionless interbank market to examine liquidity
runs. While the model suggests that pure liquidity shocks do not provide for a complete
rationalization for deposit insurance, other justi�cations for deposit insurance may be
produced in this framework when additional frictions are added. For example, if an
individual bank has large real asset losses, it will fail due to fundamental insolvency,
which will trigger a run. Deposit insurance may be helpful for an orderly liquidation.
9Bryant (1980), Calomiris and Kahn (1991), Chang and Velasco (2000), Freixas et al. (2000, 2004),Freixas and Holthausen (2005), Gale and Vives (2002), Peck and Shell (2003), Postlewaite and Vives(1987), Repullo (2000) and Rochet and Vives (2004).10Martin (2005) shows that central bank lending in a real deposits model describes historical commodity
money rather than modern �at money as in the nominal deposits model of Martin (2006), which followsAllen and Gale (1998).
4
However, failures of banks are special relative to other �rms because of their fragile liability
structure. In order to distinguish the real e¤ects of asset solvency shocks from the liquidity
e¤ects of liability shocks� i.e. excessive depositor withdrawals� the liquidity e¤ect of
nominal contracts and interbank monetary payments to lessen or compound runs must be
understood. For example, Skeie (2004) shows that with nominal deposits, systematic real
asset losses do not lead to bank runs; but a coordination failure in the interbank market
causes not only bank runs but also contagion, which requires a lender of last resort rather
than deposit insurance to resolve. These results with nominal contracts are due to a
nominal monetary price mechanism, which is not present in real deposit models. Thus, it
is important that real deposit models of bank runs that are triggered by asset shocks and
asymmetric-information should also be re-examined in a nominal deposits and interbank
payments framework.
This paper relates particularly to a few other works. Allen and Gale (1998, 2000b) add
nominal contracts through central bank loans of currency, which the bank pays depositors
in addition to goods, in order to de�ate the real value of deposit withdrawals during an
equilibrium bank run. The currency is stored outside of the banking system by withdrawers
until a later period, when they use it to purchase goods from the bank. The bank repays
the currency to the central bank and pays remaining depositors in real goods for their
withdrawals. In the current study, nominal deposits are contracted ex-ante to repay only
in money, so that during a potential run prices would rise and real deposit values would
fall endogenously, which prevents bank runs in equilibrium.11
Diamond and Rajan (2006) and Champ et al. (1996) examine bank runs with nominal
contracts and money in general equilibrium.12 They are explicit in modeling how bank
runs result if there is a large enough demand for currency withdrawals out of the banking
system. Diamond and Rajan (2006) also examine the bank�s asset side and show that
nominal contracts cannot prevent bank runs caused by idiosyncratic delays in asset returns.
I examine the bank�s liability side and show that nominal contracts can prevent bank runs
caused by depositor liquidity-driven withdrawals.
Jacklin (1987) shows that depositors purchasing assets with real bank deposits destroy
11Skeie (2004) shows that with the model of nominal deposits in the current study, prices would respondto the aggregate asset shocks in Allen and Gale (1998) to achieve their optimal consumption allocationwithout the bank runs or central bank injections of currency that occur in Allen and Gale (1998). Thisalso suggests that nominal contracts as modeled in the current study would prevent runs in the case ofaggregate shocks in the fraction of consumers who fundamentally need to withdraw early.12See also Boyd et at. (2004a, 2004b).
5
the bank�s optimal risk sharing. I show a very di¤erent point that depositors purchas-
ing current goods with nominal bank deposits achieve the bank�s optimal risk sharing.
Jacklin (1987) also shows that an equity market can replicate the optimal risk sharing
of a bank that issues real deposits, without bank runs. I show that a bank that issues
nominal deposits can also replicate the optimal risk sharing of a bank that issues real
deposits, without bank runs. This paper also relates to the literature on clearinghouse
payment systems and interbank markets, by developing a model of a clearinghouse in
which monetary payments are driven by and important for real economic activity.13
Section 2 presents and gives results for the underlying real model. Section 3 introduces
the nominal deposit model. Section 4 shows how prices in the goods market can prevent
runs with a single representative bank. Section 5 develops the clearinghouse model and
shows how the interbank market can prevent runs with multiple banks. Section 6 extends
the results from a centralized goods market to a sequential goods market, in which price
dynamics re�ect the discovery of a potential bank run over time. Section 7 discusses the
results, and Section 8 concludes with potential implications regarding deposit insurance.
2 Real Goods Model
2.1 The Environment
The environment of the real model, without money and entrepreneurs, is that which
has become standard in the literature based on Diamond and Dybvig (1983). There are
three periods, t = f0; 1; 2g: A continuum of ex-ante identical consumers with unit mass is
endowed with a unit mass of a good at t = 0: At t = 1; a fraction � 2 (0; 1) of consumers
receive an unveri�able liquidity shock and need to consume in that period. These �early�
consumers have utility given by U = u (c1) ; where c1 is their consumption in t = 1.
The remaining fraction 1 � � are �late�consumers. They have utility U = u (c2) ; where
c2 is their consumption in t = 2. Late consumers can store any goods they receive
in t = 1 for consumption in t = 2: Consumption ct is expressed as goods per unit-
sized consumer. The allocation consumed by early and late consumers is expressed as
(c1; c2). Period utility functions u(�) are assumed to be twice continuously di¤erentiable,
increasing, strictly concave and satisfy Inada conditions u0(0) = 1 and u0(1) = 0: I
13See for example Rochet and Tirole (1996), Flannery (1996), Freeman (1996), Green (1999a, 1999b),Henckel et al. (1999) and McAndrews and Roberds (1995, 1999).
6
make the typical assumption following Diamond-Dybvig that the consumers�coe¢ cient of
relative risk aversion is greater than one, which implies that banks provide risk-decreasing
insurance against liquidity shocks.
At t = f0; 1g; any fraction of goods can be stored for a return of one in the following
period. At t = 0; any fraction of goods can alternatively be invested for a return of r > 1
at t = 2: These invested goods can be liquidated for a salvage return of s < 1 at t = 1 and
zero return at t = 2, which re�ects the ine¢ ciency of liquidation.
2.2 Optimal Allocation
The (ex-ante) optimal allocation for consumers is what a benevolent planner could provide
based on observing consumer types to maximize a consumer�s expected utility. This
establishes a benchmark to compare against banking allocations. The planner�s problem
is:
maxc1;c2;�;
�u(c1) + (1� �)u(c2)
s.t. �c1 � 1� �+ s
(1� �) c2 � (�� )r + 1� �+ s� �c1;
where � � 1 is the fraction of goods that are invested at t = 0; and � � is the amount
of goods invested at t = 0 that are liquidated at t = 1. The �rst constraint says that
early consumers can only consume from goods stored at t = 0 plus invested goods that
are liquidated at t = 1: The second constraint says that late consumers can only consume
from returns of invested goods that are not liquidated and current goods available at
t = 1 that are not consumed by early consumers. Optimal consumption requires that
early consumers only consume from goods stored at t = 0, and that late consumers only
consume from the returns of invested goods. This ensures no ine¢ cient liquidation and
no underinvestment of goods. The �rst-order conditions and binding constraints give the
7
solution as the optimal allocation (c�1; c�2) and optimal choice of �
� and �; de�ned by
u0(c�1)u0(c�2)
= r (1)
�c�1 = 1� �� (2)
(1� �) c�2 = ��r (3)
� = 0:
Equation (1) shows that the ratio of marginal utilities between t = 1 and t = 2 is equal
to the marginal rate of transformation r:
2.3 Spot-Market Solution
A spot market in which invested goods are traded for current goods by consumers at t = 1
is not able to achieve the optimal allocation, a well-known result that generates the role
for a bank. In a spot market, consumers invest � goods and store 1�� goods at t = 0: At
t = 1; � early consumers trade their � invested goods for the 1� � stored goods of 1� �
late consumers at price pinv; expressed as current goods per invested good. In equilibrium,
pinv =(1��)(1��)
�� = 1: After trading in the market, consumption for early and late types
is
c1 = 1� �+ �
pinv= 1
c2 = [�+ (1� �) pinv]r = r:
The assumption of the coe¢ cient of relative risk aversion greater than one implies c�1 > 1
and c�2 < r;14 so the market allocation does not provide enough goods to early consumers
at t = 1: Since pinv = 1 implies 1�� = �; whereas c�1 > 1 implies from (2) that 1��� > �;
the amount of stored goods is less than optimal: 1� � < 1� ��:
Optimal consumption for early consumers requires in e¤ect an insurance payo¤ against
their early consumption shock. However, standard insurance cannot be provided since
types are not veri�able. Neither would consumers store enough goods at t = 0 to provide
this insurance through the spot market, which would require late consumers at t = 1 to
pay a higher price than pinv = 1 for the invested goods sold by the early types at t = 1:
14The coe¢ cient of relative risk aversion �cu00(c)u0(c) > 1 implies that cu0(c) is decreasing in c: Hence,
u0(1) > ru0(r); so from (1), c�1 > 1, c�2 < r.
8
Rather, if pinv > 1 were expected, all consumers would invest even more goods at t = 0
to sell at high t = 1 prices, which would imply pinv < 1: Thus, this is not an equilibrium.
I provide a novel result that shows the market does provide optimal consumption if
consumers are forced to store the optimal amount of goods at t = 0: This is important
because it suggests that a key role of a bank is to ensure su¢ cient amounts of liquid goods,
and with this provision markets can distribute goods e¢ ciently. If consumers were forced
to store 1� �� goods; the market achieves the optimal outcome. Early consumers would
trade �� invested goods for 1��� stored goods from late consumers at t = 1 at a market
price of pinv =c�1c�2r > 1:
Proposition 1. The allocation in the unique market equilibrium when consumers are
required to store 1� �� and invest �� is the consumers�optimal outcome (c�1; c�2):
Proof. See Appendix.
2.4 Banking Solution with Real Goods Deposits
A bank can o¤er demand deposits repayable in real goods to provide the optimal allocation
for consumers as an equilibrium outcome, which is the seminal Diamond-Dybvig result.
Consumers deposit their goods with the bank at t = 0 for a real goods demand deposit
contract. The deposit contract pays a return of either c�1 or c�2 in goods on demand
to a depositor who withdraws at either t = 1 or t = 2; respectively. Withdrawals are
payable according to a sequential service constraint, or �rst-come �rst-served until the
bank�s current and invested goods are depleted. The bank is owned by its depositors
for the purpose of maximizing their t = 0 expected utility. At t = 0; the bank stores
1��� goods and invests �� goods. Since the �rst-order condition (1) implies c�2 > c�1; the
incentive constraint for consumers to truthfully reveal their type is satis�ed in the �good�
equilibrium. Early consumers withdraw at t = 1 and late consumers withdraw at t = 2.
A bank run is a possible equilibrium as well, however. If at t = 1 late consumers
believe that all other late consumers are going to withdraw early at t = 1 from the bank,
it is a best response for each to withdraw early and a bank run occurs. Real-goods demand
deposit contracts require the bank to liquidate investments for excess withdrawals at t = 1;
causing a default on repayments by the bank and a sub-optimal allocation for consumers.
Deposit insurance guarantees that late consumers are as well o¤ by withdrawing at t = 2
9
as at t = 1; even if there is a run. Late consumers do not withdraw early, so deposit
insurance implies a bank run is no longer an equilibrium.
3 Money and Nominal Deposits Model
I extend the real banking model such that demand deposits are repayable in money. To
combine money and banking, I introduce a simple model of an �ideal banking system�as
proposed by Wicksell.15 A bank holds nominal deposits and loans. All payments in the
economy are made internal to a single bank in this section and made within a clearinghouse
system of multiple banks in Section 5. For simplicity, no reserves are held by banks, which
implies an in�nite money multiplier on deposits. This corresponds to a system of free
banking, or laissez-faire, with the assumptions of no currency payments and no frictions
within the interbank payment and lending system. As Wicksell suggests, reserves are not
fundamentally necessary if payments (his �medium of exchange�) are made in a unit of
account that is nominal, which corresponds to his �measure of value�of money divorced
from metal (or goods).
I also extend the real banking model such that the real storage and investment of goods
are done by entrepreneurs outside of the bank. A unit continuum of entrepreneurs take
loans from the bank to purchase goods, which the entrepreneurs store and invest. They
sell proceeds on a goods market at t 2 f1; 2g to repay the loan. Entrepreneurs have no
endowment, are risk neutral and maximize pro�ts in terms of unsold goods they consume
15�If customers are in direct business contact the money need never leave the bank at all, but paymentcan be made by a simple transfer from one banking account to another... If we suppose for the sake ofsimplicity that all such business is concentrated in a single bank... All payments will be made by chequesdrawn on the payer�s banking account, but these cheques will never lead to any withdrawals of money fromthe bank, but only to a transfer to the payee�s account in the books of the bank... The lending operationsof the bank will consist rather in its entering in its books a �ctitious deposit equal to the amount of theloan, on which the borrower may draw... [P]ayments...must naturally lead to a credit with another person�s(seller�s) account, either in the form of a deposit paid in or of a repayment of a debt...��The ideal banking system sketched above has in recent times engaged the attention of many writers...and
various proposals for its realization have been made. That developments tend in this direction is clear. Weneed only look at the English, German, and American banks with their �clearing house�... Theoretically thisimaginary system is of extraordinary interest... Some authors, both earlier and more recent have leaned tothe view...that mere �bank cover�, i.e. the holding of bills and securities in the portfolios of the banks as thesole basis of note issues and cheques would be the ideal... Indeed, our modern monetary system is a ictedby an imperfection, an inherent contradiction. The development of credit aims at rendering the holding ofcash reserves unnecessary, and yet these cash reserves are a necessary, though far from su¢ cient guaranteeof the stability of money values... Only by completely divorcing the value of money from metal...and bymaking...the unit employed in the accounts of the credit institutions, both the medium of exchange and themeasure of value� only in this way can the contradiction be overcome and the imperfection be remedied.�Knut Wicksell (1906), Lectures on Political Economy, Volume Two: Money
10
at the end of t = 2. Due to competition, I assume entrepreneurs are subject to a zero
pro�t condition. Without loss of generality, I treat the continuum of entrepreneurs as a
single competitive entrepreneur who is a price taker.
3.1 Nominal Unit of Account
At t = 0; the bank issues loans and takes deposits denominated in a nominal unit of
account, which is created using the following ad-hoc technique. A central bank exists only
at t = 0: It creates �at currency with which it stands ready to buy and (to the extent
feasible) sell goods at a �xed price of P0 = 1. The central bank receives all currency back
by the end of the t = 0 period and plays no role thereafter. Even though the supply of
central bank money is zero after t = 0, the role of central bank currency as a nominal unit
of account carries over to later periods due to monetary contracts established at t = 0.16
As illustrated in Figure 1, consumers sell their one unit of goods to the central bank for
one unit of central bank currency at price P0 = 1: The consumers deposit their currency
I0 = 1 in the bank in exchange for a nominal demand deposit contract. The deposit
contract pays a return of either D1 or D217 in monetary payments on demand when a
depositor withdraws at either t = 1 or t = 2; respectively, subject to a sequential service
constraint. The bank sets
D1 = c�1 (4)
D2 = c�2: (5)
The bank lends the unit of central bank currency to the entrepreneur for the nominal
loan contract repayments of
K1 = 1� � (6)
K2 = �r; (7)
due at t = 1 and t = 2; respectively; where Kt is payable in monetary payments. The
entrepreneur buys the good from the central bank with its currency. The bank sets � = ��
16A similar process for money sequentially taking on an abstract unit of account, which is ultimatelybased on an original commodity value, is described by Kitson (1895), pages 6-8, and von Mises (1912/1971),pages 108-123.17Uppercase letters denote variables with nominal values and lowercase letters continue to denote vari-
ables with real values.
11
CentralBank
EntrepreneurConsumers
Bank
1 good
$1
$1
$1
$1
1 good
LoanContractK1 and K2
DepositContract(D1, D2)
CentralBank
EntrepreneurConsumers
Bank
1 good
$1
$1
$1
$1
1 good
LoanContractK1 and K2
DepositContract(D1, D2)
Figure 1: Introduction of Nominal Contracts at t = 0. A central bank buys andsells goods with �at currency at a �xed price of P0 = 1 to establish nominal contracts.Consumers sell their good to the central bank for currency. They deposit the currency in thebank for a nominal demand deposit contract (D1; D2), which repays Dt upon withdrawal ateither t = 1 or t = 2. The bank lends the currency to the entrepreneur for the nominal loancontract repayments of K1 and K2 due at t = 1 and t = 2; respectively : The entrepreneurbuys the good from the central bank with the currency.
and can ensure that the entrepreneur stores 1�� = 1��� and invests � = �� at t = 0: This
is a key function of a bank, since as suggested above, the market allocation of consumption
is optimal once su¢ cient storage is enforced.
At the end of this exchange at t = 0; the net holdings are as follows. The central bank
holds all the currency and does not hold any goods. The consumer holds the demand
deposit account that repays Dt: The entrepreneur holds 1�� stored goods and � invested
goods. The bank holds the loan contracts that repay K1 and K2 from the entrepreneur.18
3.2 Timeline
A bank run in Diamond-Dybvig can be interpreted as a fear either that the bank will
default on withdrawals at t = 2; or that long term investments will be liquidated and
depleted, leaving no consumption goods available at t = 2: These interpretations are
indistinguishable, because bank deposits are repayable in consumption goods. These in-
terpretations are made clear in this paper by the distinction of the withdrawal of deposits
from the purchase of consumption goods. If late depositors fear their bank will default on
withdrawal payments at t = 2; they can withdraw money to redeposit at a di¤erent bank
at t = 1: I call this a �redeposit run,�and examine it in Section 5 with a multiple-bank
18Money and nominal contracts could be introduced without the central bank actually exchanging goodsfor currency, but with just the guarantee to do so at t = 0, which still establishes the nominal unit ofaccount. Consumers could deposit goods directly with the bank for the nominal demand deposit accountof Dt and the bank would lend the goods to entrepreneurs for K1 and K2:
12
model in which depositors can withdraw either to redeposit or to purchase goods. If late
consumers fear that a bank run will lead to the liquidation of investments and the deple-
tion of consumption goods from the goods market, they can withdraw money to purchase
goods at t = 1: I call this a �purchase run,�and examine it �rst with a single-bank model
in which depositors can withdraw only to purchase goods.
In the single-bank model, monetary payments are made by debiting and/or crediting
accounts internal to the bank. Depositor withdrawals at t = 1 or t = 2 are made by
debiting the depositor�s account and crediting the entrepreneur�s account at the bank.
These withdrawals are recorded as a demand schedule with a Walrasian auctioneer, and
the entrepreneur simultaneously submits a supply schedule qSt to the auctioneer. The
entrepreneur then repays Kt if possible, which debits the entrepreneur�s account and
credits the bank�s account. If the entrepreneur�s account nets out to be negative, he
defaults at period t and must liquidate and sell all goods possible, de�ned as bqSt ; inattempt to repay his loan. If the entrepreneur has positive net credits in his account at
t = 1; the are held as deposits. I assume the bank pays a return on deposits made at t = 1
of
D1;2 =D2D1;
because this is the endogenously derived return on deposits made at t = 1 in Section 5
with multiple banks.19 Since c�1 > 1 and c�2 < r; equations (4) and (5) imply
D1;2 < r:
The fraction of depositors withdrawing at t = 1 is de�ned as �p; where �p � � and
the superscript �p�denotes these withdrawals as �purchases.� I assume that late consumers
do not withdraw at t = 1 if they are indi¤erent. A bank run equilibrium in the single-
bank model is called a �purchase run�and is de�ned as �p > �: Since all payments are
made internal to the bank, the bank cannot default. The sequential service constraint is
inconsequential in this section, but it is applied in the multiple-bank model in Section 5
where a bank default is possible. Although the bank would not default in this section if
there were a purchase run, the optimal consumer allocation would not obtain. Note that
the consumers�optimal allocation (c�1; c�2) and the e¢ cient �
� and � are the same as in
the real deposits banking model.
19Payments and accounts are modeled more formally in the clearinghouse model of Section 5.
13
Market clearing and price determination occurs simultaneously with calculating inter-
nal accounts at the bank to determine if the entrepreneur defaults. A competitive market
equilibrium for goods is de�ned as a solution (Pt; qt) for t 2 f1; 2g that solves
P1q1(P1) � �pD1 (8)
P2q2(P2) � (1� �p)D2; (9)
where Pt is the price, qt = qSt is the goods clearing equilibrium condition and qSt solves the
entrepreneur�s optimization problem given below. Prices equate the value of goods sold,
Ptqt(Pt); to the quantity of money spent on goods, �pD1 at t = 1 and (1��p)D1 at t = 2.
Solving for prices in (8) and (9),
P1 = �pD1q1
(10)
P2 = (1��p)D2q2
: (11)
The prices in a period is equal to money spent divided by goods for sale, corresponding to
a simple version of the quantity theory of money. After market clearing, qt(Pt) is delivered
to depositors purchasing goods.
In the case of a bank run by all late depositors, �p = 1: The entrepreneur does not sell
any goods at t = 2 because if he did, P2 would be zero. Hence, P2(�p = 1) = 00 is unde�ned.
Without loss of generality, I de�ne P 12 � P2(�p = 1); and assume P 12 2 (0;1).20
4 Single-Bank Results
Here I examine the entrepreneur�s and consumers� problems. The entrepreneur�s opti-
mization problem is to maximize his pro�t subject to repaying his loan, which determines
the relative amount of goods sold in each period. I �rst show that there are no purchase
runs. Then the zero pro�t condition is imposed to determine prices and I show the optimal
consumer allocation obtains.20 If any zero-mass set of late depositors were to withdraw at t = 2; spending a zero-mass amount
of money to buy goods, the entrepreneur would be indi¤erent to selling them any zero-mass amount ofgoods. The resulting price could be any positive, �nite amount. The proof of Lemma 2 below shows thatfor �p = 1; any P 12 2 (0;1) is consistent with the entrepreneur�s choice of qS2 = 0 such that the �rst orderconditions of the entrepreneur�s optimization hold.
14
4.1 Entrepreneur Optimization
The entrepreneur chooses qS1 (P1) and qS2 (P2); subject to repayingK1 andK2, taking prices
as given. The market clearing equilibrium condition qt = qSt for t 2 f1; 2g is then imposed.
First, I show the entrepreneur never defaults. The entrepreneur can always repay K1
because it is set equal to the amount early consumers withdraw �D1 to purchase goods,
seen from (6), (2) and (4). Consider the case of qS1 > 0: Substituting the equilibrium
condition q1 = qS1 in the de�nition of P1 and rearranging, the qS1 P1 revenues received by
the entrepreneur at t = 1 equal �pD1 � K1: Next consider the case of qS1 = 0: This implies
P1 = 1; which means money is worthless and consumers are indi¤erent between paying
money or not in the market since they receive no goods regardless. I assume that when
consumers are indi¤erent they do not pay money, so the entrepreneur has to sell qS1 > 0
to not default.21
The entrepreneur can always repay K2 as well. K2 is set equal to the amount of
withdrawals that late consumers spend on goods (1� �)D2 if they all withdraw at t = 2;
seen from (7), (3) and (5). If late consumers withdraw any money at t = 1; it is spent
on goods at t = 1 and ends up as excess revenues that the entrepreneur receives and
holds as deposits. The entrepreneur receives on this sum the return D1;2 that any early-
withdrawing late consumers implicitly forgo. The entrepreneur�s revenues at t = 2 total
qS2 P2 + (qS1 P1 �K1)D1;2 = (1� �p)D2 + (�p � �)D1D1;2 (12)
= K2:
On the LHS, the �rst term is the entrepreneur�s revenues from t = 2 sales of goods, and
the second term is the entrepreneur�s return on t = 1 deposits. Imposing the equilibrium
condition qt = qSt and substituting for prices from (10) and (11) gives the RHS of (12),
which equals K2:
Equation (12) implies qS1 and qS2 cannot be independently chosen by the entrepreneur.
Rearranging the equation, the choice of qS1 determines
qS2 =K2+D1;2K1
P2� qS1D1;2 P1P2 (13)
21Formally, qSt PtjqSt =0 � 0 for t 2 f1; 2g:
15
that is required to repay K2: The entrepreneur�s pro�t at t = 2 is
bqS2 � qS2 ; (14)
where bqS2 is the entrepreneur�s total available goods at t = 2; equal to the goods from
storage, liquidation and investment returns minus the goods sold at t = 1:
bqS2 � 1� �+ s+ (�� )r � qS1 : (15)
The entrepreneur�s optimization can be considered as his choice of how many goods to
sell at t = 1 relative to t = 2 to maximize pro�ts. Speci�cally, the entrepreneur chooses
qS1 and : After substituting for bqS2 and qS2 from (15) and (13) into (14), the maximization
of the entrepreneur�s pro�t (14) can be written as
maxqS1 ;
[1� �+ �r � (r � s)� qS1 (1�D1;2 P1P2 )�K2+D1;2K1
P2j �p] (16a)
s.t. qS1 � 1� �+ s (16b)
� �; (16c)
with the requirement that qS1 and are nonnegative. The �rst constraint (16b) says thatbqS1 � 1� �+ s is the most goods that can be sold at t = 1: The second constraint (16c)says that � is the most invested goods that can be liquidated at t = 1: Since P1 and P2
re�ect �p; the entrepreneur�s maximization problem is written as conditional on �p: I can
now formally prove that the entrepreneur never defaults.
Lemma 1. The entrepreneur never defaults on repaying K1 and K2:
Proof. See Appendix.
Next, I examine the �rst order condition from the entrepreneur�s optimization, which
gives insight into how market prices prevent a purchase run. The entrepreneur�s choices
imply that P1 is always greater or equal to discounted P2 regardless of �p:
The �rst order condition (40) is restated from the proof of Lemma 1:
D1;2P1P2= 1 + �1; (17)
where �1 is the nonnegative Lagrange multiplier associated with the feasibility constraint
16
(16b) on qS1 . One plus �1 represents the t = 2 shadow value of selling a marginal good at
t = 1 relative to at t = 2. The �rst order condition can be written in terms of marginal
revenues. Since �1 � 0;
P1 �P2D1;2
: (18)
The entrepreneur attempts to equate discounted marginal revenues from sales at t = 1 and
at t = 2: The LHS of (18) is the entrepreneur�s marginal revenue P1 of selling additional
goods at t = 1; whereas the RHS is the entrepreneur�s marginal revenue P2 discounted to
t = 1 by D1;2 of selling additional goods at t = 2: The inequality is due to the asymmetry
between selling an additional good at t = 1 versus at t = 2: The discounted t = 2 price
of goods is never greater than the t = 1 price of goods. If discounted P2 were greater
than P1; the entrepreneur would store goods over from t = 1 to sell at t = 2: An increase
(decrease) in q1 (q2) would drive P1 (P2) higher (lower), until P1 = P2D1;2
:
Conversely, suppose �p were to increase due to a purchase run, driving up P1 above
discounted P2: The entrepreneur would have to liquidate investments to sell more goods
at t = 1: The entrepreneur�s �rst order condition under > 0 is:
sP1 = rP2D1;2
: (19)
The LHS (RHS) gives the discounted marginal revenues from selling an additional good
at t = 1 (t = 2): P1 is greater than discounted P2 by the marginal rate of transformationrs from not liquidating an invested good, implying (18) does not bind if > 0:
4.2 Consumer Optimization
The late consumers�problem is to choose whether to withdraw and purchase goods at
t = 1 or t = 2: To analyze a late consumer�s decision when to withdraw, I compare his
real consumption DtPtfrom withdrawing and purchasing goods at t = 1 versus at t = 2:
A late consumer is better o¤ withdrawing and purchasing goods at t = 1 if and only if
D1P1
>D2P2: (20)
However, the entrepreneur�s �rst order condition (18) can be expressed as D1P1� D2
P2; so
(20) never holds. Hence, a late consumer never wants to run the bank to purchase goods.
17
Proposition 2. There is no purchase run in the unique Nash equilibrium of the late
consumers�problem of the single-bank model, given by �p = �; which is an equilibrium in
dominant strategies.
Proof. See Appendix.
To explain, a late consumer only wants to run the bank if, rewriting (20), P1 < P2D1;2
;
or P1 is low relative to P2: This would mean that a relative abundance of goods is for sale
at t = 1: But if P1 is low, the entrepreneur would shift selling goods from t = 1 to t = 2;
driving P1 up until P1 � P2D1;2
: The late consumer never has to run the bank at t = 1 to
purchase goods because the market always provides goods for patient depositors at t = 2.
Thus the no purchase run result is a unique equilibrium. Moreover, the preference by the
late consumer to withdraw and purchase goods at t = 2 according to D2P2� D1
P1holds by
(18) for all �p: The no purchase run outcome is an equilibrium in dominant strategies.
Furthermore, I do not need to make any assumptions regarding the late consumers�beliefs
or symmetry of actions.
This is in important contrast to Diamond-Dybvig. Banks are fragile in Diamond-
Dybvig because the no bank run outcome is a Nash equilibrium that depends on coor-
dinated beliefs that other late consumers do not run. A shift in beliefs that other late
consumers will run triggers the bank run equilibrium. In Diamond-Dybvig, the greater
the number of late consumers that run the bank, the less the bank can pay out in goods
at t = 2; and so the greater the desire for a marginal late consumer to run. In this paper,
since goods are sold by the market, a late consumer prefers to withdraw at t = 2 even
if other late consumers run the bank. In fact, the greater the number of late consumers
who run the bank, the higher is P1 (lower is P2) and the lower is consumption D1P1at t = 1
(higher is consumption D2P2at t = 2): The price mechanism rations goods to depositors
who run, ensuring an even greater desire of a marginal late consumer not to run.
4.3 Optimal Allocation
It remains to be shown that the allocation for consumers is the optimal one of (c�1; c�2) from
the planner�s problem. The relative quantities sold by the entrepreneur in each period were
solved for above. Now I �nd the actual quantities that the entrepreneur sells. The zero
pro�t condition for the entrepreneur imposes that the entrepreneur pro�t bqS2 � qS2 equalszero in equilibrium.
18
The optimal consumer allocation is achieved if there is no liquidation of invested goods
at t = 1 and no storage of current goods from t = 1 to t = 2. This means that all current
goods are sold at t = 1; q1 = 1� �; and all returns from invested goods are sold at t = 2;
q2 = �r: By substituting these quantities into prices in (10) and (11), where �p = �, prices
in the optimal outcome are P1 = P2 = 1:
To see that the optimal allocation is achieved by the market, �rst consider the contrary
case that the entrepreneur would store current goods at t = 1: The entrepreneur would
never liquidate invested goods if he were storing goods at t = 1, so = 0: Also, the
entrepreneur would sell all current goods at t = 1 if P1 > P2D1;2
; so storage of current goods
at t = 1 implies (18) binds. Restated, storage at t = 1 implies that the constraint on qS1 ;
(16b), does not bind, so �1 = 0 in (17) and
P1 =P2D1;2
: (21)
Storage at t = 1 implies a decrease of q1 and an increase of q2; driving prices from the
optimal outcome of P1 = P2 = 1 to P1 > 1; P2 < 1: But this contradicts (21): marginal
discounted revenues are not equated between t = 1 and t = 2: This shows that the
entrepreneur does not store goods at t = 1.
Second, consider the contrary case of liquidation of invested goods. This implies the
entrepreneur is subject to the �rst order condition (19). Liquidation implies an increase
of q1 and a decrease of q2; driving prices from the optimal outcome of P1 = P2 = 1 to
P1 < 1; P2 > 1: In (19), the LHS marginal revenues from liquidating and selling goods
at t = 1 are less than s; while the RHS discounted marginal revenues from selling returns
from the invested goods at t = 2 are greater than one, since D1;2 < r: This contradicts
(19): marginal discounted revenues are not equated between t = 1 and t = 2: Thus, the
entrepreneur does not liquidate invested goods.
Proposition 3. The allocation in the unique market equilibrium of the single-bank model
is the consumers�optimal consumption (c�1; c�2).
Proof. See Appendix.
This result also implies that nominal deposits are a Pareto optimal improvement over
real deposits. Consumers receive optimal consumption with no risk of bank runs when they
hold nominal deposits. If consumers were to hold real deposits, they would be exposed
19
to the risk of bank runs and suboptimal consumption. If a second bank were to compete
for deposits at t = 0 by o¤ering real demand deposits against the original bank o¤ering
nominal demand deposits, consumers would all deposit at the original bank and nominal
deposits would be the unique equilibrium.
With the assumption of a zero pro�t condition imposed ex-post on the entrepreneur
in this model, prices are determined and the consumers�optimal allocation obtains. The
robustness of this assumption is studied in the Appendix. Dropping this assumption allows
for potential price indeterminacy, but this does not e¤ect the result of no bank runs in the
single-bank model or the multiple-bank model below. Rather, di¤erent price equilibria
can lead to transfers between consumers and the entrepreneur. These equilibria are not
Pareto-ranked because there are no ex-ante or ex-post ine¢ ciencies. In the Appendix, I
consider a zero-pro�t condition assumed ex-ante rather than imposed ex-post. The bank
can ensure that prices are determined by adjusting the entrepreneur�s loan repayments,
which results in the consumers�ex-ante optimal allocation. Alternatively, the application
of a stable equilibrium concept introduced in the Appendix achieves the same result.
5 Clearinghouse in a Multiple-Bank Model
5.1 Clearinghouse Model
In this section, I extend the single-bank model to a multiple-bank model. This allows
for examining �redeposit runs,� in which depositors run their bank by withdrawing and
redepositing at a di¤erent bank. A second bank, labelled �bank B�, allows for t = 1
payments and redeposits to accounts outside of the original bank, now labeled �bank A�.22
At t = 1; early consumers withdraw to purchase goods. Late consumers may now with-
draw early either to purchase goods or to redeposit funds to bank B. I de�ne �w � �p as
the fraction of consumers who withdraw at t = 1; where the superscript �w�denotes �total
withdrawals�at t = 1: The fraction �w��p of consumers withdraw and redeposit at bank
B. A bank run equilibrium is rede�ned as �w > �, which can be due to either a purchase
run, �p > �; or due to a �redeposit run,�de�ned as �w > �p: Bank B o¤ers return DB2 on
deposits IB2 made at t = 1 by redepositors and/or the entrepreneur. Bank B is owned by
22Bank B may represent multiple banks and for simplicity does not o¤er deposits and loans itself att = 0: If it did, it would also have the potential for runs with its late consumers purchasing goods orredepositing to bank A. The results are unchanged, in that interbank lending implies that no bank wouldfail and there would be no runs in equilibrium.
20
its t = 1 depositors and acts to maximize their t = 1 utility. For simplicity, I assume bank
A does not take new deposits from the entrepreneur at t = 1: The entrepreneur receives
and makes payments and holds deposits at an account with bank B at t 2 f1; 2g: If late
consumers are indi¤erent, they keep their deposits at bank A.
A clearinghouse budget constraint for each bank in period t 2 f1; 2g is based on a
�payment-in-the-same-period�constraint, in contrast to the traditional �cash-in-advance�
constraint. Payments are made between banks A and B within a clearinghouse under
deferred net settlement, explained as follows.23 A clearinghouse is an intraperiod �balance
sheet�with accounts for banks A and B. Banks A and B also have internal accounts for
depositors and the entrepreneur, separate from the clearinghouse accounts. At the start of
t 2 f1; 2g, both banks start at zero balances. A payment debits the paying bank�s account
and credits the receiving bank�s account, and additionally debits or credits internal bank
accounts as appropriate. A bank can make any amount of payments during a period and
so carry a negative intraperiod clearinghouse balance. Payment credits and debits are
provisional until settled at the end of the period.
At t 2 f1; 2g; depositor withdrawals for redeposits and purchases are paid from bank
A to bank B. Consumers with deposits at bank B withdraw at t = 2 to purchase goods, so
bank B only adjusts internal accounts. Withdrawals to purchase goods are also recorded as
a demand schedule with a Walrasian auctioneer. The entrepreneur simultaneously submits
a supply schedule qSt to the auctioneer. The entrepreneur repays Kt on his loan to bank
A. At t = 1; Bank B makes a take-it-or-leave-it loan o¤er of an amount F � 0 at a return
of DF2 (corresponding to the U.S. federal funds rate) to bank A. Bank A chooses action
� 2 f0; 1g, where � = 1 (� = 0) corresponds to �accept�(�reject�). If bank A accepts, it
repays DF2 F at t = 2:
At the end of the period, a bank with a net negative balance owes this amount in
central bank currency to the other bank. Since banks have no currency, they must �nish
with a zero net balance for their provisional payments to �settle� and take e¤ect. If a
bank�s payments do not settle, only the largest partial amount of its payments, calculated
�rst according to the seniority of the claim, and second according to the sequential order
in which they were made during the period, that do net out to zero are called �settled.�
23While this model represents a private clearinghouse, such as CHIPS under its former deferred netsettlement system, the model could be adjusted to represent a real-time gross settlement clearinghouseoperated by a central bank, such as the Federal Reserve�s Fedwire, without changing the results.
21
All payments that do not settle are unwound and defaulted upon by the bank.24
At the end of the period the following occurs simultaneously: clearing in all markets
and the determination of prices, netting and settlement of payments through the clear-
inghouse and calculating internal accounts at banks A and B. The market for monetary
balances held overnight by banks clears through the price DF2 chosen by bank B at t = 1
in (30) below: The market for goods clears through the price Pt in t 2 f1; 2g determined
by the market equilibrium conditions (24) and (25) below. Bank B�s budget constraint to
settle payments is given by (29), (31) and (32) below. First, I examine bank A�s budget
constraint to settle payments in each period:
t = 1 : �w�1D1 � �F + �1K1 + �1;2K2 (22)
t = 2 : (1� �w)�2D2 + �F�F2 DF2 � �2(1� �1;2)K2: (23)
De�ne �t 2 [0; 1] as the fraction that settles on the entrepreneur�s loan repayment of Ktin period t: If �t < 1; the entrepreneur defaults and must sell all goods, including any
invested goods that must �rst be liquidated. The variable �it 2 [0; 1] is de�ned as the
fraction that actually settles of the contracted return Dit due, where i 2 fA;B; Fg and the
notation i = A is suppressed to conform with the single-bank model. If �it < 1; the bank
paying Dit defaults at period t: If bank A were to default at t = 1; it liquidates its long
term loan to the entrepreneur only as a last resort by calling enough of it for repayment
to try not to default. De�ne �1;2 2 [0; 1] as the fraction of K2 called by bank A that is
due to be repaid at t = 1 and settles. This leaves (1� �1;2)K2 due at t = 2: This implies
that �1 < 1 only if �1;2 = 1: Furthermore, �1;2 > 0 implies � = 1:
In (22), the RHS loan from bank B and the loan repayment from the entrepreneur
that settles to bank A at t = 1 must be greater than or equal to the LHS deposits
repaid and settled by bank A to early withdrawers. In (23), the RHS loan repayment
from the entrepreneur that settles to bank A at t = 2 must be greater than or equal to
the LHS deposits repaid and settled by bank A to late withdrawers. If a bank defaults,
the amount of withdrawal payments that do settle is �it; where i 2 fA;B; Fg: Because the
clearinghouse rules give settlement to payments in the order they were made, the sequential
service constraint holds. Interbank loans have a junior claim to demand deposits, which
24The model abstracts away from risk within the payments system, since there is no risk to the receivingbank if a payment does not settle. Payments risk within a clearinghouse is an important issue and isstudied in Rochet and Tirole (1996).
22
implies �2 < 1 only if �F2 = 0:25
A competitive market equilibrium for goods is de�ned as in the single-bank model,
with the exception that (8) and (9) are replaced by
P1q1(P1) � �p�1D1 (24)
P2q2(P2) � (1� �w)�2D2 + (�w � �p)�1D1�B2 DB2 ; (25)
where qSt solves the entrepreneur�s optimization. Prices that clear the goods market are
now solved as
P1 =�p�1D1q1
(26)
P2 =(1� �w)�2D2 + (�w � �p)�1D1�B2 DB2
q2; (27)
where the numerators re�ect the settled payments for purchasing goods by consumers
at each period. The settled amount the entrepreneur repays on its loan at t = 1 is
�1K1 + �1;2K2; where �1 � minf�D1;�p�1D1gK1
; and at t = 2 is �2(1� �1;2)K2; where
�2 �minf(1� �1;2)K2; (1� �w)�2D2 + [(�1�w � �1�)D1 � �1;2K2]�B2 DB2 g
(1� �1;2)K2:
The entrepreneur delivers qt(Pt) goods at period t 2 f1; 2g to purchasers whose payments
settle.
5.2 Multiple-Bank Results
I �rst examine the interbank borrowing and lending problems of the banks. In order to
maximize the return on its t = 1 deposits, bank B would always prefer to lend any funds it
has to bank A to capture part of the return bank A receives from its long term investments.
Bank A can borrow fully from bank B and so does not default even if there were extensive
early withdrawals. Then I examine the entrepreneur�s and consumers�problems. Since
bank A never defaults, the optimizations collapse to those from the single bank model.
25This is necessary so that the second bank cannot expropriate late consumers who do not withdrawat t = 1 when it lends to the original bank, and so it is a natural condition of the demand depositcontract that the original bank includes at t = 0 in order to maximize t = 0 depositors�welfare. The U.S.created a national depositor-preference law giving depositors priority on the assets of failed banks overother claimants as part of the Omnibus Budget Reconciliation Act of 1993 following the FDICIA depositinsurance reforms in 1991.
23
There are no redeposit runs or purchase runs, and the optimal allocation obtains.
5.2.1 Interbank Lending
Bank A�s problem is to attempt not to default. The bank has to repay deposits at t 2
f1; 2g: It chooses whether to accept an interbank loan and then has to repay that as well.
The bank�s optimization is
max�;�1;�2
�1(�) + �2(�)
s.t. (22)
(23).
Since �1 < 1 implies �2 = 0; bank A chooses �; �1 and �2 based on maximizing payouts
�rst to early withdrawers through �1; and second to late withdrawers, through �2; subject
to its budget constraints (22) and (23).
Bank B�s problem is to maximize the return it pays its t = 1 depositors. Deposits
received by bank B at t = 1 are
IB2 = (�w � �p)�1D1 + P1q1 � �1K1 � �1;2K2 (29)
= (�1�w � �1�)D1 � �1;2K2
� 0:
The �rst term on the RHS of (29) is the quantity of redeposits received from late consumers
withdrawing at t = 1. The remaining terms are the quantity of deposits received from the
entrepreneur, equal to the excess of the entrepreneur�s revenues in the �rst period beyond
his loan repayment. Since there is no further uncertainty, bank B�s optimization is to
24
maximize the return �B2 DB2 it pays depositors:
maxF;DF
2 ;�B2 D
B2
�B2 DB2 (30)
s.t. �F � IB2 (31)
IB2 �B2 D
B2 � �F�F2 DF2 (32)
�F2 DF2 � 1 (33)
(22)
(23).
Equations (31) and (32) are Bank B�s budget constraints at t = 1 and t = 2, respectively.
The maximum return that bank B can repay on deposits is given from (32) by
�B2 DB2 �
�F�F2 DF2
IB2: (34)
Equation (33) is the rationality constraint for bank B to lend. Equations (31) and (34)
show that bank B prefers to lend out as much as possible of the deposits it receives, at the
highest return that bank A will accept and be able to pay and settle, given by ��F2 DF2 ;
in order to pass on the maximum return it can to its depositors. Equations (22) and (23)
are Bank A�s budget constraints, which constrain the amount that bank B can collect as
return on its loan.
Bank A always accepts the loan because if F > 0; bank A defaults without it. The
only amount bank A needs to borrow is the F = �pD1 � �D1 o¤ered by bank B. This is
the amount that late consumers withdraw at t = 1 for redeposits and purchases that end
up deposited at bank B by the redepositors and entrepreneur. Bank A can always pay a
return up to D2D1on the loan it needs since this is the implicit rate between t = 1 and t = 2
that it would have paid to late consumers if they withdrew at t = 2 instead of earlier.
Bank A cannot pay more since it just breaks even when paying that rate. If bank B were
to o¤er DF2 >D2D1; bank A would accept but just default to bank B on the excess, resulting
in �F2 DF2 =
D2D1: This would not e¤ect any bank A depositors withdrawing at t = 2 since
interbank loans are junior to deposits.26 Thus, DB2 = DF1;2 =D2D1
is the deposit rate at
26 If interbank loans were senior to deposits, bank B would accept redeposits from late consumers and thendemand �F2 D
F2 > D2
D1and extort remaining bank A late consumers. All late consumers would redeposit,
implying a full run and failure of bank A. This occurs due to bank B�s bargaining power. If the interbank
25
t = 1:27
Lemma 2. Bank B lends F = (�w��)D1 to bank A at a return of DF2 = DB2 = D2D1: There
are no defaults by bank A, bank B or the entrepreneur and no calls on the entrepreneur�s
loan:
�F2 = �1 = �2 = �B2 = �1 = �2 = 1
�1;2 = 0:
Proof. See Appendix.
5.2.2 Entrepreneur and Consumer Optimizations
The entrepreneur�s optimization problem and the goods market outcome are identical to
those in the single-bank model. Since there are no defaults and DB2 = D1;2 =D2D1; prices
given by (26) and (27) are unchanged from prices given in the single-bank model by (10)
and (11), respectively. If there is a redeposit run, bank A pays the implicit return D2D1;
which redepositors forgo, instead as a return on the interbank loan to bank B, which
pays it to the redepositors. Thus, the amount of money spent on purchases at t = 2 is
una¤ected by redeposits. Hence, the entrepreneur�s maximization problem is identical to
(16) in the single-bank problem. The entrepreneur�s �rst order condition (18) continues
to hold.
The late consumers�optimization is expanded to choose whether to run the bank at
t = 1, and either redeposit or purchase goods; or to withdraw and purchase goods at
t = 2: I compare the real consumption from the three options. A late consumer is better
o¤ withdrawing and redepositing at t = 1 if and only if it is preferred to purchasing goods
at t = 2;�1D1�
B2 D
B2
P2>�2D2P2
; (35)
market were dispersed among many banks who were price-takers as consumers are, bank A would nothave runs. To the extent the interbank market can act strategically, the seniority of demand deposits isimportant.27Bank B would choose to lend fully to bank A even if lending to new entrepreneurs with new investment
projects were possible. Lending to bank A is optimal for bank B and social welfare since it provides thegreatest possible one-period nominal return of D2
D1to bank B and real return of r
sto society from continuing
the invested goods instead of liquidating them.
26
and preferred to purchasing goods at t = 1; �1D1�B2 D
B2
P2> �1D1
P1: A late consumer is better
o¤ withdrawing and purchasing goods at t = 1 if and only if it is preferred to purchasing
goods at t = 2;�1D1P1
>�2D2P2
; (36)
and preferred to redepositing at t = 1,
�1D1P1
>�1D1�
B2 D
B2
P2: (37)
Bank B cannot pay a higher return on deposits between t = 1 and t = 2 than bank A, so
DB2 =D2D1and the LHS of (35) equals D2P2 : Bank A never defaults, so �2 = 1 and the RHS
of (35) equals D2P2 : Hence, (35) does not hold, so late consumers never withdraw early to
redeposit at bank B.
This is again in contrast to Diamond-Dybvig. In that paper, the more late consumers
who run the bank, the likelier it will fail and the more that a marginal late consumer
prefers to run. In the current study, the early withdrawals by others do not impact the
ability of a marginal late consumer to withdraw at t = 2: Thus, there are no possible
beliefs to trigger a redeposit run.
No defaults and DB2 =D2D1imply that the late consumer�s criteria to withdraw at t = 1
to purchase goods, (36) and (37), collapse to (20), the criteria in the single-bank model.
From the single-bank model, the entrepreneur�s �rst order condition (18) implies (20) does
not hold, so the late consumers do not withdraw early to purchase goods.
Proposition 4. There is no bank run in the unique Nash equilibrium of the late con-
sumers�problem of the multiple-bank model, given by �w = �p = �, which is an equilibrium
in dominant strategies. The allocation in the unique market equilibrium is the consumers�
optimal consumption (c�1; c�2).
Proof. See Appendix.
6 Sequential Market
An important reason why real demand deposit contracts allow bank runs in Diamond-
Dybvig is that the contract is not contingent on the realized state of �p: The assumption
of a sequential-service constraint implies that the deposit contract cannot depend on �p
27
as the �rst depositors withdraw at t = 1 because �p is not known until �p depositors
have already withdrawn. Thus, demand deposits repay a non-contingent �xed amount of
consumption goods c�1 to each depositor who withdraws at t = 1 until all goods are paid
out.
In the nominal contracts model, nominal payouts by the bank are also not contingent
on �p: The �xed payout (in money) of the deposit contract and the sequential service
constraint are strictly adhered to. However, real consumption �1tD1P1
= q1�p to the depositor
withdrawing at t = 1 is contingent on �p through P1: This is because the market for
goods is assumed to be a centralized one-price Walrasian market. The price in the goods
market is based on aggregate withdrawals at t = 1 and is not determined until �p is
realized. Although the payout by the bank does not settle until �p and P1 are realized, the
sequential service constraint is formally upheld by the bank because the monetary amount
paid and settled on a withdrawal for a depositor does not depend on the withdrawals after
him. The payment from the bank for a depositor withdrawal settles in full before the
payment for the next depositor withdrawal settles.
However, this may not re�ect the spirit of the sequential service constraint as inter-
preted by Wallace (1988). He requires that a withdrawer receives and consumes his real
consumption before the next depositor withdraws, due to an assumption that consumers
are physically isolated from each other and a liquidity shock at t = 1means that consumers
need to consume immediately within the period rather than at the end of the period.
6.1 Sequential Market Model
The Wallace (1988) requirement for what I call his �sequential consumption constraint�
may be easily met in the nominal contracts model with the introduction of a sequential
market that replaces the centralized market in the multiple-bank model as follows. In
t = 1; depositors in a random order sequentially discover their early or late type and have
the opportunity to sequentially withdraw and consume in that order. A consumer who
discovers being an early type needs to withdraw and consume before the next consumer
in the sequential order discovers his type and has an opportunity to withdraw. The late
consumer�s consumption c2 is modi�ed to equal goods he consumes in both t = 1 and
t = 2:
Before a withdrawal, the competitive entrepreneur posts a price Pt(�p), where �
pis
the cumulative fraction of depositors who have purchased goods up to that point within
28
period t: Bank B also posts a deposit rate DB2 . Then a single depositor may withdraw to
purchase goods or to redeposit. Bank A makes a payment to bank B for the withdrawal
amount. If there is a purchase, the payment goes to the entrepreneur�s account at bank
B. The entrepreneur uses the funds to pay a portion of K1 to bank A if the loan is
outstanding, or else keeps the funds on deposit with bank B at DB2 . Bank B may lend to
bank A. Markets clear and clearinghouse payments are netted out and settled. Bank A�s
payment must settle before the bank makes a payment for the next depositor withdrawal,
otherwise it defaults. If a purchase payment does settle, the entrepreneur delivers goods to
the purchaser according to P1(�p) and the depositor immediately consumes. The process
repeats with the entrepreneur and bank B posting a new P1(�p) and DB2 ; respectively, and
the next depositor discovering his type and choosing whether to withdraw. The realization
of �p does not take place until the last depositor decides whether to withdraw at t = 1;
by which point all other earlier purchasers have consumed their purchases. At t = 2; late
consumers who have not yet purchased goods make purchase withdrawals from their bank
and consume in a sequential random order in a similar fashion to withdrawals in t = 1.
6.2 Entrepreneur and Consumer Optimizations
The outcome in this sequential market model is the same as that of the centralized market
model in Section 5. The budget constraint and optimization problems for banks A and B
are unchanged from the centralized market model in Section 5. The sequential payments
for purchases, redeposits, entrepreneur loan repayments and interbank loans and repay-
ments aggregate up in each period t 2 f1; 2g and are equal to the equivalent lump-sum
aggregate payments in the centralized market model. Bank B lends incrementally any
amount needed by bank A at a rate of D2D1 : The entrepreneur repays its loan, and bank
A does not default on deposits or interbank loans. This implies late consumers do not
redeposit, so there is no redeposit run.
The question of if there is a purchase run is more subtle. Whether �p = � or �p > �
is expected at the start of t = 1; the entrepreneur�s optimization problem is unchanged
from the centralized market model. If a purchase run �p > � is expected, the entrepreneur
applies the �rst order condition under liquidation (19), sP1 = r P1D1;2
; to prices at t = 1.
29
Rearranging, this implies
D1P1
=sD2rP2
<D2P2;
a contradiction of the late consumers�optimization (20). Late consumers do not prefer to
purchase goods early, so �p = � and there is no purchase run.
If �p = � is expected, the entrepreneur sets P1(�p= �) = P2(�
p = �) = 1; and
sells the 1 � � stored goods to the �rst � purchasers at t = 1: If additional purchasers
appear after the �rst � fraction, the entrepreneur posts a new P1(�p> �) to equalize
discounted revenues between a marginal amount of invested goods sold (i) in liquidation
for sP1(�p> �) at t = 1; versus (ii) at full return for rP2(�
p> �) at t = 2. Equation (19)
still holds, but with P1(�p> �) substituted in for P1 as the price for marginal goods sold
to a marginal purchaser at t = 1: The new sequential market equilibrium equation that
de�nes P1(�p> �) and replaces (26) is
�qS1P1(�p> �) � ��pD1;
where ��pis the size of the marginal purchaser and �qS1 = � s is the marginal goods
sold from liquidating � marginal invested goods. Since �p is known at the start of t = 2;
the de�nition of P2 (27) is unchanged and the entrepreneur posts a �xed P2 throughout
t = 2:
Excess early purchases at t = 1 do not end up changing P2 from its value when there
is no run: P2(�p> �) = P2(�
p = �) = 1: This is because the loss from liquidating
invested goods is fully borne by early purchasers who buy the liquidated invested goods.
Substituting P2 = 1 in (19),
P1(�p> �) =
r
sD1;2> 1:
The consumption for these early purchasers is sD1;2r ; compared with consumption for
purchasers at t = 2 of D2: Since P1(�p> �) > P2 implies (20) does not hold, no late
consumer would choose to purchase goods at t = 1 after the �rst � fraction of depositors
30
have purchased goods. Moreover, a late consumer who purchases goods among the �rst
� fraction at P1(�p � �) = 1 is also worse o¤ than purchasing goods at P2 = 1 at t = 2;
according to (20). This is because late purchasers�consumption at t = 2 is dependent on
P2 = 1 and so is una¤ected by whether other late consumers run at t = 1: Thus, there is
no purchase run.
Proposition 5. There is no bank run in the unique Nash equilibrium of the late con-
sumers�problem of the multiple-bank model with sequential markets, given by �w = �p = �,
which is an equilibrium in dominant strategies. The allocation in the unique market equi-
librium is the consumers�optimal consumption (c�1; c�2).
Proof. See Appendix.
The real consumption of each withdrawer at t = 1 does not need to depend on the
number of early withdrawers following him, as revealed in �p by a centralized market price,
for runs to be avoided. Rather, the sequential market ensures that the real consumption
of each early withdrawer depends on only the number of withdrawers before him to ration
excess early consumption and dissuade a run. Although the price for each transaction at
t = 1 is set based on perfect information of the number of prior purchasers, this could be
relaxed. Withdrawers at t = 1 could each purchase goods at t = 1 from a random draw of
an entrepreneur out of a continuum of informationally isolated entrepreneurs rather that
from a single entrepreneur. Each individual entrepreneur would charge P1(�p> �) > 1
for any liquidated goods he sells and P2 = 1 for t = 2 sales. Thus, a late consumer would
always be better o¤ withdrawing at t = 2:
7 Discussion
Demand deposits repayable in money achieve the no bank runs result, whereas deposits
repayable in real goods allow for runs in Diamond-Dybvig. To interpret this result, I
discuss the four features of money as payment in the model that prevents runs. Money is
i) in the form of a liability claim, ii) which is denominated in a nominal unit of account,
iii) the claim is a liability of the bank making the payment, and iv) is payable to banks
who can coordinate lending.
Payment in the form of a real or nominal claim prevents redeposit runs. Instead, if
payment is in goods, an interbank lending market with multiple banks does not su¢ ce to
31
prevent a redeposit run. Late consumers could withdraw goods from bank A at t = 1 and
redeposit at bank B, but bank A would �rst have to liquidate invested goods to repay late
consumers their withdrawal. Liquidation costs imply that bank A would default even if
bank B were to lend to bank A. Thus bank B would not lend and redeposit runs could
occur in equilibrium. Deposits repayable with claims on real goods, in which the claim is
a liability of bank A and is redeemable on demand under a sequential service constraint,
prevent redeposit runs. Late consumers could withdraw claims at t = 1 and redeposit
them at bank B. Bank B would lend the claims back to bank A, and bank A could repay
bank B in goods at t = 2: There would be no liquidation or default, so a run never starts.
However, real claims are not su¢ cient to eliminate all runs. If depositors believe
that everyone is withdrawing and redeeming claims at t = 1; bank A has to liquidate
invested goods and will default. If real claims were the liability of the entrepreneur, a
run for redemption on the entrepreneur also leads to liquidation and default. Claims
denominated in a real unit of account have a �xed real value.28
Claims denominated in a nominal unit of account that are redeemable in the nom-
inal numeraire by the end of the period, settled on a netting basis, have a �exible real
value. This value is determined by prices in a market of goods for claims, which prevents
purchases runs. In this paper, central bank currency is the nominal unit of account.29
If deposits were indexed to prices, deposits would have a real value instead of a nominal
value and purchase runs could occur. Alternatively, if the central bank continued to set a
�xed price in t 2 f1; 2g, money would have a real unit of account and purchase runs again
could occur. For example, a central bank that sets a gold standard or pegs its currency
could induce a run, in which a bank paying withdrawals in domestic money does not fail,
but the central bank loses all reserves and has to devalue the currency.
The �nal features of money to prevent runs is that bank A pays a claim to bank B
for depositor withdrawals, and the claim is a liability on bank A itself, which is �inside
28As in Diamond-Dybvig, in the centralized market model I take as given that bank deposits are paid ondemand under a sequential service constraint. In the environment of the sequential market model in thispaper, deposits paid on demand under a sequential service constraint are optimal features of the depositcontract, as shown by Wallace (1988).29 I do not examine whether the clearinghouse could establish the nominal unit of account at t = 0 instead
of a central bank. This question is related to whether a central bank or clearinghouse is necessary to achievee¢ ciency in payments, examined in Green (1999a). Without either a central bank or clearinghouse unitof account, a competing bank paying inside money in its own self-de�ned nominal unit of account couldoverissue deposits and in�ate money payments. I also do not address whether settlement should occurat the end of each period or only at the end of t = 2: Koeppl et al. (2006) study a similar question ofsettlement frequency in a search model of payments.
32
money,� as opposed to a liability on the central bank, which is currency or �outside
money.�When the bank pays inside money, it pays a liability that the bank must settle
in currency by the end of the period. Bank A can pay any amount of claims on itself for
depositor withdrawals on demand, i.e. it has a perfectly elastic supply of its own liabilities,
since the present value of its liabilities equals that of its assets. These claims are paid for
withdrawals from bank A for purchases or redeposits through the clearinghouse to bank
B, which can lend them all back. Thus, the liability to pay currency at the end of the
period nets out to zero.
Because bank A pays claims for withdrawals to bank B on behalf of depositors, bank
B resolves the coordination problem of late depositors running bank A. However, Skeie
(2004) shows that a bank run could occur instead due to a coordination problem of lend-
ing in the interbank market, if multiple banks that are informationally dispersed receive
clearinghouse payments from bank A. Alternatively, a run could occur if the claim with-
drawn on deposits is currency rather than inside money, due to depositor withdrawals
and storage of currency outside of the banking system. A remedy for either run is for the
central bank to lend currency, since the central bank can ensure an elastic supply of its
own liabilities.
With no currency held in the model at t 2 f1; 2g; there is no need for creating an ad-
hoc demand for holding money, which is required in other papers when money otherwise
pays a lower return than other assets. Typical assumptions include a cash-in-advance
constraint and the �scal theory of money, which are both applied by Diamond and Rajan
(2006); a loan injection and then extraction of currency by the central bank, which is
applied by Allen and Gale (1998); and money in the utility function, used by Chang and
Velasco (2000).
In this paper, at the beginning and end of each period t 2 f1; 2g; bank A�s interperiod
�balance sheet� liabilities (deposits and interbank loans) and assets (entrepreneur loans)
are equal to each other on a present value basis. Any intraperiod monetary payments
bank A makes are equal to payments received and net out. There is no need in the model
for a supply of currency to be held between periods, which would receive a zero return, to
use for making payments within periods.
The assumption of no outside money reserves in the model is abstract and intended to
be parsimonious, to focus on the central issue of modern interbank payments and lending.
But the model may also describe fairly well the very small size of bank reserves and net
33
settlement payments relative to the large size of bank deposits and gross payments in
modern economies in practice. For example, banks have no reserve requirements in the
U.K., Canada, Australia, New Zealand and Sweden, and hold very low reserves.30 In 2004,
Canadian banks held $50 million in central bank reserves, only 0.02% of the $270 billion
held of demand deposits,31 while New Zealand banks held $20 million in central bank
reserves, only 0.1% of the $20 billion held of demand deposits.32 Even in the U.S., banks
held $24 billion in central bank reserves,33 only 0.5% of the $5.2 trillion held of total bank
deposits.34 From their reserves of $24 billion, U.S. banks made on average $1.9 trillion
in gross payments per day through the Federal Reserve�s clearinghouse system Fedwire,35
equivalent to a 79 times turnover ratio in reserves per day. Banks made an additional $1.2
trillion in gross payments per day through the private clearinghouse CHIPS (in 1995),
which after netting required only $7 billion, or 0.6% of gross payments, in actual transfers
of bank reserves per day to achieve settlement.36 In this context, the model assumptions
of zero reserves and zero transfers of reserves for settlement of payments approximates the
very small percentages of deposits and gross payments that these values take in practice.
8 Conclusion
The Federal Deposit Insurance Corporation (FDIC) has been widely credited for the dras-
tic reduction in the occurrence of bank runs in the U.S. since it was established in 1933.
However, FDIC only covers deposits up to a limited size, which is greatly exceeded by
many large depositors. This paper can suggest an additional explanation for the decrease
in bank runs. Nearly all of the theoretical bank run literature examines real demand de-
posits and describes historical bank runs based on currency withdrawals from the banking
system. This paper adds to the Diamond-Dybvig model the features found in current
practice of nominal deposits payable through a clearinghouse for monetary payments, in-
terbank markets, and entrepreneurs taking loans and selling goods in a market outside of
30Woodford (2000).31Amounts are Canadian dollars. Sources: Bank of Canada (2005) and International Monetary Fund
(2005), respectively.32Amounts are New Zealand dollars. Sources: Reserve Bank of New Zealand (2004) and International
Monetary Fund (2005), respectively.33Board of Governors of the Federal Reserve System (2005b).34Board of Governors of the Federal Reserve System (2005a).35Board of Governors of the Federal Reserve System (2006).36Richards (1995).
34
the banking system. Together, these features in a modern economy suggest that interbank
lending and monetary market prices may be able to prevent pure liquidity runs.
First, �exible prices due to unbacked currency and a �oating exchange rate in the U.S.
in recent times imply that nominal contracts allow prices for goods to vary with the level
of withdrawals, ensuring no purchase runs. Previous to FDIC, the value of money was tied
to gold so deposits more resembled real contracts. Second, modern clearinghouses with
instant electronic payments since the 1960s and 1970s imply that money is not depleted
from the banking system when payments are made between banks, so interbank lending
allows money to be e¢ ciently distributed, ensuring no redeposit runs. In earlier times,
currency withdrawals likely were the fastest, most con�dent method of withdrawing funds,
given the risk of delays in transferring funds to another bank and the risk that a bank
would not be able to borrow during the same day from other banks if depositors withdrew
excessively.
This paper highlights how Diamond-Dybvig may continue to justify deposit insurance
in developing countries with �xed exchange rates or bank deposits repayable in foreign
currency. However, the Diamond-Dybvig justi�cation for deposit insurance does not nec-
essarily hold for developed countries. Demirguc-Kunt et al. (2005) show that deposit
insurance is more likely to be adopted by generally more developed countries, where I
argue it may be less needed, than in generally less developed economies, where it may be
more needed, under the hypothesis that more developed countries adopt due to bene�ts
to private interests. Given the evidence cited in the introduction that deposit insurance
may increase bank failures due to moral hazard, but bene�t private interests, this paper
may lend support to the argument questioning whether deposit insurance may hurt more
than help and may be adopted for rent-seeking reasons.
These resulting policy conclusions must be quali�ed because asset shocks are not exam-
ined in the current model. Large asset losses that threaten a bank�s fundamental solvency
would cause bank runs, so if this leads to ine¢ cient liquidation then deposit insurance
may be bene�cial to provide for an orderly bank closure without liquidation. Examining
this, along with asymmetric information in the interbank market, in a nominal contracts
and interbank payments framework as presented here is needed to more fully understand
nominal versus real e¤ects and is left for further research. For example, Skeie (2004)
shows that with nominal contracts, banks are hedged against aggregate real asset losses,
but information breakdowns at the interbank level can lead to liquidity runs and conta-
35
gion. Analysis on crises at the interbank level shows that deposit insurance cannot help,
and instead a lender of last resort is necessary. Additional next steps for research include
applying nominal contracts to study �nancial stability issues incorporating bank reserves,
capital requirements and clearinghouse payment risks.
36
Appendix A: Robustness
Price Indeterminacy I drop the zero pro�t condition that is imposed ex-post for the
entrepreneur to examine the issue of price indeterminacy and the relationship between
ex-ante and ex-post entrepreneur pro�t. I continue to assume the entrepreneur is a com-
petitive price taker and that he maximizes revenue by choosing to sell goods in t = 1
versus t = 2: However, the assumption that the bank o¤ers the entrepreneur debt con-
tracts K1 = 1 � � and K2 = �r allows for the market equilibrium of zero entrepreneur
pro�t as well as additional market equilibria that transfer allocations from consumers to
the entrepreneur for positive entrepreneur pro�t. The price level at t 2 f1; 2g and alloca-
tion of consumption is indeterminate. While the relative price level P2P1 is determined, the
indeterminacy of the absolute level of prices is a standard problem in general equilibrium
theory. Here the indeterminacy e¤ects real allocations between consumers and the entre-
preneur�s pro�ts. Since the entrepreneur receives PtqSt = Kt at t 2 f1; 2g for all qt > 0;
he can always repay his loan, but Pt and qt are not determined. Outcomes of lower con-
sumption for consumers and positive pro�t for the entrepreneur are not ine¢ cient because
they are not Pareto-ranked. Rather they are simply transfers to the entrepreneur. The
�rst order condition for the entrepreneur in (17) can be expressed as
q2 2 [(1� �)D1; (1� �)D2] if q1 = �D1 (38)
q2 =
�1� ��
�q1 if q1 < �D1: (39)
If q1 = �D1; P1 = 1: Any P2 2 (1; D1;2] corresponding to q2 2 [(1� �)D1; (1� �)D2)
in (38) implies the entrepreneur pro�ts at the late consumers� expense. However, late
consumers do not run according to (20) and the entrepreneur�s �rst order condition (18),
which holds since discounted P2 is still lower than P1: If q1 < �D1; P1 and P2 are greater
than one, implying the entrepreneur pro�ts at the early and late consumers�expense. The
entrepreneur�s �rst order condition (18) is binding and there is no run according to (20).
The possibility for positive entrepreneur pro�t and price indeterminacy may be resolved
by the bank o¤ering di¤erent loan repayment contracts. If there is any positive probability
at t = 0 of an equilibrium with P2 > 1 in which the entrepreneur pro�ts under the original
K1 and K2 contracts, the entrepreneur has an ex-ante expected positive pro�t. The bank
can o¤er debt contracts of K1 = 1 � � and K2 = �r + "; where " > 0: K2 > �r implies
37
the entrepreneur always defaults at t = 2; which requires that qS2 = bqS2 , so Pt = 1 and theconsumers�allocation is (c�1; c
�2) as the unique equilibrium. Even though the entrepreneur
always defaults in equilibrium, due to limited liability the entrepreneur makes zero pro�t.
However, the zero pro�t condition is imposed her ex-ante instead of ex-post.
Stable Market Equilibrium Price determinacy may also be resolved without requiring
the entrepreneur to default in equilibrium using a stable market equilibrium concept.
The zero entrepreneur pro�t allocation is the unique outcome and prices are uniquely
determined without a zero pro�t condition imposed ex-post.
A stable market equilibrium is de�ned as the limiting case of an �-equilibrium in which
� 2 (0; 1 � �) late consumers do not withdraw in any period. An �-model is equivalent
to the multiple-bank model with the following modi�cations. A fraction � consumers are
early, 1� �� � consumers are late, and � > 0 consumers die unexpectedly at the start of
t = 1 before they have a chance to withdraw. Thus, �w � 1� �.
First, I consider the single-bank model. The variables qSt (�); qt(�) and Pt(�) are sub-
stituted in the model for qSt ; qt and Pt. Equation (9) is replaced by
P2(�)q2(�) = (1� �p � �)D2:
An �-equilibrium of the �-model is de�ned as a competitive market equilibrium (Pt(�); qt(�))
and a �p Nash equilibrium of the late consumers�problem.
I de�ne a competitive market equilibrium (Pt; qt) for t 2 f1; 2g as a stable market
equilibrium if qt = lim�!0 qt(�): For � > 0; qS2 (�)P2(�) < K2; so there is not enough money
paid for goods for the entrepreneur to repay K2: The entrepreneur defaults on repaying
K2 by an amount �D2: Hence, the entrepreneur must sell all goods available qS2 (�) = bqS2at t = 2: The entrepreneur�s �rst order condition (17) is unchanged in the �-equilibrium.
It is shown in the proof below that this implies he sells all goods qS1 (�) = bqS1 = 1� �+ sat t = 1 as well. The late consumer�s condition for early purchases (20) contradicts
(18), so late consumers withdraw at t = 2. In the �-equilibrium, for all �; there is no
run, and (Pt(�); qt(�)) is determined as P1(�) = �D1bqS1 ; P2(�) = (1����)D2bqS2 and qt(�) = bqSt :Hence, in the stable market equilibrium, (Pt; qt) is determined as P1 = �D1bqS1 ; P2 = (1��)D2bqS2and qt = bqSt : By the entrepreneur maximizing his revenues, the market price mechanismprovides optimal goods at t 2 f1; 2g: The entrepreneur does not liquidate any invested
38
goods at t = 1 and does not store any goods from t = 1 to t = 2:
Second, I consider the multiple-bank model. The stable market equilibrium is de�ned
as in the single-bank model, with the exception that (8) and (9) are replaced by
P1(�)q1(�) = �p�1D1
P2(�)q2(�) = (1� �w � �)�2D2 + (�w � �p)�1D1�B2 DB2 :
For any �-equilibrium, the entrepreneur defaults at t = 2: He receives (1��p��)D2+(�p�
�)D1DB2 from purchasers at t = 2 and from his deposits at bank B. Since DB2 =
D2D1; the
entrepreneur�s revenues at t = 2 are less than K2 by the amount �D2: The entrepreneur
must sell qS2 (�) = bqS2 at t = 2 due to the default. The entrepreneur�s �rst order condition(17) is unchanged and implies that qS1 (�) = bqS1 :
Bank A does not default. Since it owes only (1 � �w � �)D2 on deposits at t = 2;
it owes �D2 less on deposits at t = 2 than otherwise, equal to the amount by which the
entrepreneur defaults. Bank A receives from the entrepreneur at t = 2 the amount it owes
on deposits plus FDB2 due to bank B. Thus, bank B lends fully to bank A at a rate ofD2D1: The late consumers�optimization in the multiple-bank �-model collapses to that of
the single-bank �-model, so there is no run. Since qSt (�) = bqSt is determined, the stablemarket equilibrium is determined as (Pt; bqSt ):Proposition 6. The allocation in the unique stable market equilibrium of the single-bank
model and the multiple-bank model is the consumers�ex-ante optimal consumption (c�1; c�2);
and zero entrepreneur pro�ts.
Proof. See Appendix Proofs.
39
Appendix B: Proofs
Proof of Proposition 1: The market clearing price of invested goods in terms of stored
goods at t = 1 is pinv =(1��)(1���)
��� : Consumption is given by
c1 = 1� �� + (1� �) (1� ��)
�=1� ���
= c�1
c2 = ��r +���r
1� � =��r
1� � = c�2:
The trade is incentive compatible for late consumers since the value of invested goods
received from trading is greater than the value of current goods paid,
(1� ��)rpinv
= �c�2 > �c�1 = (1� ��):
The trade is incentive compatible for early consumers if the value of liquid goods received
from trading is greater than the value of liquidating invested goods,
pinv�� =
c�1��r
c�2> s��;
which holds since the coe¢ cient of relative risk aversion greater than one implies c�2c�1< r:
Q.E.D.
Proof of Lemma 1: The nonnegative Lagrange multipliers associated with the con-
straints (16b) and (16c) are �1 and �2. The necessary Kuhn-Tucker conditions are:
D1;2P1P2
= 1 + �1 (40)
r � s+ �2 � �1s; (41)
and combining (40) and (41) gives
sD1;2P1P2� r(1 + �2); (42)
with (41) and (42) binding if > 0:
First I show that the de�nition of P 12 2 (0;1) is consistent with the entrepreneur�s
choice of qS2 : Consider �p = 1: Suppose the entrepreneur sells goods at t = 2; qS2 > 0:
This implies P2 = 0; and also that either �1 or �2 equals zero by complementary slackness,
40
which is a contradiction to either (40) or (42), respectively, since P1 > 0: Hence, qS2 = 0
and since (1� �p) = 0; P2(�p = 1) is unde�ned by (9). For any P 12 2 (0;1); there exists
q1 2 [0; 1��+ s] such that �1 = D2P2q1
� 1 � 0 and �2 = sD2rP2q1
� 1 � 0; implying (40) and
(42) hold. For illustration, if P2 is low, then > 0 and �1 and �2 are high. If P2 is high,
then q1 < 1� �; P1 is high, and �1 = �2 = 0:
I have shown in the text that K1 is always repaid for qS1 > 0: Suppose qS1 = 0:
qS1 P1jqS1=0 = 0 by de�nition, so the entrepreneur defaults. This require qS1 = bqS1 > 0; a
contradiction. Thus qS1 > 0. To show K2 is repaid, consider the case of �p = 1: This
implies qS1 P1 = D1; which after the entrepreneur repays K1 at t = 1 leaves D1 �K1 for
deposit at rate D1;2: The deposit value is (D1 �K1)D1;2 = K2 to repay K2 at t = 2:
Next consider the case of �p < 1: Suppose q2 = 0: This implies P2 = 1; so by (40),
1+�1 = 0; a contradiction to �1 � 0: Thus, q2 > 0 and q2P2 = (1��p)D2: The entrepreneur
has at t = 2 available to repay (q1P1�K1)D1;2+ q2P2 = (�p� �)D2+ (1� �)D2 = K2 so
never defaults. Q.E.D.
Proof of Proposition 2: A late consumer withdraws at t = 1 to purchase goods if and
only if (20) hold. Since (18) holds in any market equilibrium, withdrawing at t = 1 is a
strictly dominated strategy. Withdrawing at t = 2 is a dominant strategy for each late
consumer. Thus, �p = � and no purchase runs is the unique Nash equilibrium and is an
equilibrium in dominant strategies. Q.E.D.
Proof of Proposition 3: First, I prove (19). Suppose > 0 and qS1 < 1 � � + s:
This implies �1 = 0 by complementary slackness, and (41) binding implies r = s� �2 < s;
a contradiction to r > s: Thus, > 0 implies qS1 = 1 � � + s: Next, suppose = �:
Substituting P2 = 1 in (40) implies 1 + �1 = 0; a contradiction to �1 � 0: Thus, < �;
�2 = 0 due to complementary slackness and > 0 implies sP1 = r P2D1;2
.
Next, I show that q1 = bqS1 : Suppose not: qS1 < bqS1 : This implies �1 = 0; so after
substituting for prices in (17), recognizing bqS2 � (�� )r and rearranging, q1 � �p(�� )r1��p :
Suppose = 0: This implies q1 � �D2 > �D1; which implies > 0; a contradiction. Thus,
> 0 implies sP1D1;2 = rP2; but by (17), P1D1;2 = P2; implying s = r; a contradiction.
Thus qS1 = bqS1 .Finally, suppose > 0: By substituting into (19) for prices and qt = bqSt for t 2
f1; 2g: s�D1�D1+ s
= r(1��)D2D1;2(�� )r : The LHS is less than s; while the RHS is greater than one, a
contradiction to s < 1: Thus, q1 = 1� �; q2 = �r; Pt = 1 and (c1; c2) = (c�1; c�2): Q.E.D.
41
Proof of Lemma 2: First I show � = 1 in (28) and (31) binds. Since �F � IB2 =
�1�wD1 � �1K1 � �1;2K2; and (22) can be written as �F � �w�1D1 � �1K1 � �1;2K2;
(22) is binding. If �1;2 > 0; then � = 1 from the de�nition of �1;2: If �1;2 = 0; then (22)
is �F +minf�D1; �p�1D1g = �w�1D1: Since F � 0 and �p � �w; �1(�) is always weakly
increasing in �: Because �2 < 1 implies �F2 = 0; �2 does not depend on � in (23). Thus,
� = 1 maximizes �1(�) + �2(�) for all �p and �w:
Moreover, (23) binds, implying F = IB2 and (31) binds. Equation (22) can be rewritten
as �w�1D1 � �w�1D1; so bank A can choose �1 = 1: Substituting this into the de�nition
of �1; �1 = 1: Since (22) holds with �1;2 = 0; this is the solution for �1;2:
Substituting these solutions into (23) and rearranging,
�F2 DF2 �
[�2(1� �)� �2(1� �w)]D2(�w � �)D1
: (43)
I claim bank A chooses �2 = 1; which satis�es (28); bank B chooses DB2 = DF2 =
D2D1and
�B2 = 1, which satis�es (32) and (33); and show (23) and (34) are satis�ed and �F2 = 1.
Substituting �2 = 1 into (43), �F2 DF2 � D2
D1since
�2 � 1; (44)
satisfying (34). Applying these results and (32) to the de�nition of �2;
�2 =minfK2; (1� �)D2g
K2= 1: (45)
Hence, (43) and so (23) hold with equality and �F2 = 1:
Proof of Proposition 4: After substituting results from Lemma 2, the proof follows
that of Proposition 2 and Proposition 3. Q.E.D.
Proof of Proposition 5: Substituting P1(�p> �) = ��
pD1
�qS1in (19) and simplifying gives
��p= �
�1��p��
�: Integrating,
R �p� d�
p=�1��p��
� R 0 d : Solving and rearranging gives
= �(�p��)1�� ; which substituting into the de�nition of P2 gives P2 = 1: Q.E.D.
Proof of Proposition 6: First, I prove (39). If q1 < �D1; = 0 and �1 = 0 by
complementary slackness. By (19), q2 = 1��� q1: Next I prove (38). If q1 = �D1; (19)
shows q2 � (1� �)D1 since �1 � 0: The feasibility condition q2 � bqS2 implies q2 2
[(1� �)D1; (1� �)D2]:
42
Now, consider the �-model in the single bank model. Substituting in qt(�) and Pt(�);
the late consumers�optimization problem and the entrepreneur�s maximization problem
are identical. The entrepreneur does not default at t = 1 and repays (1����)D2 following
the argument from the proof for Lemma 1. The amount of default by the entrepreneur of
�D2 is the amount the bank does not repay to late consumers who die, thus the bank does
not default. Since the entrepreneur default at t = 1 does not change his maximization
problem, the �rst-order conditions given by (17), (18) and (19) continue to hold.
To show that q1 = bqS1 ; suppose qS1 < bqS1 : This implies �1 = 0; so after substituting forprices in (17), recognizing bqS2 � (� � )r and rearranging, q1 � �p(�� )r
1��p�� : Suppose = 0:
This implies q1 � �D2 > �D1; which implies > 0; a contradiction. Thus, > 0 implies
sP1D1;2 = rP2; but by (17), P1D1;2 = P2; implying s = r; a contradiction. Thus qS1 = bqS1 .Next, I show that prices and outcomes are determined. Substituting the de�nitions of
Pt(�) and bqSt (�) into (19) and rearranging, = s�p��(1��)(1��p��)s�p+s(1��p��) when it exists, otherwise
= 0: Either way, , bqSt (�); Pt(�) and ct(�) = DtPtare determined.
Lastly, consider the multiple-bank model. To examine the �-model in the multiple-
bank model, substitute for Pt(�) and qt(�) as de�ned in the single-bank model, substitute
for
�2(�) �minf(1� �1;2)K2; (1� �w � �)�2D2 + [(�1�w � �1�)D1 � �1;2K2]�B2 DB2
(1� �1;2)K2
and replace (23) with
t = 2 : (1� �w � �)�2D2 + �F�F2 DF2 � �2(1� �1;2)K2:
The argument and proof for Lemma 2 holds with the exception that (43) is replaced with
�F2 DF2 �
[�2(1� �)� �2(1� �w � �)]D2(�w � �)D1
;
(44) is replaced with �2 � 1����1�� and (45) is replaced with �2 =
minfK2;(1����)D2gK2
= 1����1�� :
Thus, there is full lending by bank B, �1;2 = 0 and there are no defaults with the exception
that �2 < 1: The remaining argument and proof for the entrepreneur�s and late consumers�
optimizations in the �-model of the multiple-bank model are identical to that of the �-model
of the single bank model. Q.E.D.
43
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